# Homework Help: Not Homework. Proving question

1. Oct 18, 2011

### icystrike

1. The problem statement, all variables and given/known data

Prove that:

$$(\frac{10^{n}}{10^{n}-1})^{10^n} = e$$

as n approaches inf.

2. Relevant equations

It is rather obvious that i can let 10^{n} be yet another variable

3. The attempt at a solution

I proved by assuming binomial dist as Poisson dist (it is interesting for me to use this to prove a continued fraction)

2. Oct 18, 2011

### Staff: Mentor

You can simplify by letting u = 10n and working with this limit:

$$\lim_{u \to \infty}\left( \frac{u}{u - 1} \right)^u$$

The usual approach is to let y = (u/(u - 1))u, and then take the natural log of both sides.

ln y = u ln(u/(u - 1)) = [ln(u/(u - 1))]/(1/u)

Now take the limit of both sides. Note that the expression above is the limit of the log of what you want.